EA3_mechsystems_1

EA3_mechsystems_1 - EA3: Systems Dynamics Mechanical...

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EA3: Systems Dynamics Mechanical Systems Sridhar Krishnaswamy 1 VI SYSTEMS DYNAMICS – MECHANICAL SYSTEMS VI.1 Modeling Mechanical Systems: We have already looked at some simple one-dimensional mechanical systems involving springs and blocks: Spring constant K (N/m) mass m r x Frictionless, No gravity (a) mass m r x Frictionless; no gravity Spring constant K 1 Spring constant K 2 (b) Figure 6.1 : (a) The undamped spring-block oscillator of Chapter III; and (b) a spring-block-spring device. In our analyses of these systems we have assumed that the block is rigid and has mass, and the spring is massless but is deformable. This is what we called a lumped parameter idealization where all the “springiness” (deformability) is attributed to the spring and all the “massiveness” is attributed to the block. In reality, real springs do have mass, and real blocks are deformable. However, ours is a useful approximation (if, for instance, the mass of the block is a lot more than that of the spring, and the stiffness of the block is a lot The course material from this point onwards draws heavily from the EA3 hyperbook of Prof. Peshkin. I will make available selected links to that hyperbook (available on the web only) at a later date. Some of

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EA3: Systems Dynamics Mechanical Systems Sridhar Krishnaswamy 2 higher than that of the spring). Also, we can actually model more complex useful real life systems with such simple idealized elements. For instance, the spring-block system might represent a heavily-loaded table or a unicycle (view Fig. 6.1a rotated CCW 90 o ). Of course more complex mechanical systems might have to be modeled with several elements. The system might have elements that not only “store” energy (potential energy in springs) and have inertia (blocks), but there may also be elements that “lose” energy (dampers) which we shall see shortly. And a real system will more often than not be two- or three- dimensional, and may have components that rotate, and not just translate in one-dimension. For simplicity, however, we will now consider mechanical systems that can only translate (no rotation) in one-dimension. Figure 6.2 shows one such system with several springs and blocks. This might represent a part of a high-rise building where the springs might represent the supporting columns, and the blocks represent the mass of the floors, all appropriately “lumped” at the locations shown. Figure 6.2 : A 12mass-13-spring system (courtesy: EA3 hyperbook) In trying to look at the dynamics of such systems, it is convenient to use a formal systems dynamics approach. In this approach, we consider the system to be comprised of several elements each with one defining property. These elements are connected to one another only at their end points, which we shall call nodes or connections . In EA2 you used a similar approach (recall the finite-element analyses of trusses?), but there you looked only at static problems (and so the mass of the various elements played no significant role
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This note was uploaded on 04/17/2008 for the course GEN_ENG 203 taught by Professor Krishnaswamy during the Fall '08 term at Northwestern.

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EA3_mechsystems_1 - EA3: Systems Dynamics Mechanical...

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